Balance for a time piece

ABSTRACT

A balance for a timepiece in which the cross-sectional shape of the balance rim is fixed so that the cross-sectional area and the perimeter of such area are related and definable in terms of a single common variable. Fixed relationship between this common variable and the mean radius of the balance rim is established by virtue of the initial instantaneous power available to oscillate the balance at the desired initial amplitude and frequency. Thus, for each value of the mean radius there exists not only a corresponding value of the cross-sectional area of the balance rim but also a corresponding value of the perimeter of such area. Optimum balance dimensions are obtained from these corresponding values by selecting that value of mean radius which, for the corresponding value of cross-sectional area of the balance rim, produces a maximum value of moment of inertia for the balance with a decrease in amplitude after 24 hours within predetermined limits.

United States Patent Simon-Vermot et al.

[ 1 July 18,1972

1541 BALANCE FOR A TIME PIECE [72] inventors: Andre Simon-Vermot, Fougeres; Charles- Andre Grossenbacher, Georges-Ferrenoud, both of Switzerland Les Fabriques dAssortiments Reunies, Neuchatel, Switzerland [73] Assignee:

[22] Filed: Sept. 1, 1970 [21] Appl. No.: 68,696

Related U.S. Applicatlon Data [63] Continuation-impart of Ser. No. 733,416, May 3l,

1968, abandoned.

301,863 12/1954 Switzerland ..58/107 339,868 9/1959 Switzerland ....58/107 356,719 10/1961 Switzerland ..58/107 Primary Examiner-Richard B. Wilkinson Assistant Examiner-Stanley A. Wal Attorney-Imirie and Smiley [5 7] ABSTRACT A balance for a timepiece in which the cross-sectional shape of the balance rim is fixed so that the cross-sectional area and the perimeter of such area are related and definable in terms of a single common variable. Fixed relationship between this common variable and the mean radius of the balance rim is established by virtue of the initial instantaneous power available to oscillate the balance at the desired initial amplitude and frequency. Thus, for each value of the mean radius there exists not only a corresponding value of the cross-sectional area of the balance rim but also a corresponding value of the perimeter of such area. Optimum balance dimensions are obtained from these corresponding values by selecting that value of mean radius which, for the corresponding value of cross-sectional area of the balance rim, produces a maximum value of moment of inertia for the balance with a decrease in amplitude after 24 hours within predetermined limits.

4 Claim, 4 Drawing Figures BALANCE FOR A TIME PIECE CROSS-REFERENCE TO RELATED APPLICATION This application is a continuation-in-part of copending application Ser. No. 733,416, filed May 31, 1968, and titled Balance for a Time Piece, now abandoned.

BACKGROUND or THE INVENTION In the past, timepiece balances have been constructed, in large part, so as to provide as large a size and consequent moment of inertia as would be allowed within the space available for the balance. The shape of the balance rim has been arbitrament of inertia. Thus, in the prior art, an increase in density of the balance or of its rim is normally employed to allow a decrease in the mean radius and/or cross-section of the balance rim. In contrast to this, the optimum conditions according to this invention demand that when the density is increased, the mean radius of the balance rim must also be increased.

BRIEF SUMMARY OF THE INVENTION In the present invention, a relationship is established between the cross-sectional area of the balance rim and the mean radius of the rim from its center of oscillation by fixing the crosssectional shape of the rim and the physical and dynamic characteristics of the balance.

The related values of mean radius and cross-sectional area of the balance rim are then, chosen so as to provide a maximum moment of inertia which, for the instantaneous power available after 24 hours and with the balance in the most unfavorable position (in a vertical plane), will not cause a decrease in oscillatory amplitude which exceeds a predetermined value (preferably 40).

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a graph illustrating a certain relationship in connection with a particular embodiment of the invention;

FIG. 2 is a plan view of a balance wheel according to the present invention;

FIG. 3 is an enlarged view partly in section illustrating certain dimensional details of one embodiment of the invention; and,

FIG. 4 is a .view similar to FIG. 3, but illustrating a modified form of the invention.

DETAILED DESCRIPTION OF THE INVENTION With reference to FIG. 2, a balance wheel is indicated generally by the reference character and will be seen to'include a hub 12, spokes 14 and a rim 16, the hub being provided with the conventional pivot shafts 18, as may be seen more clearly in FIG. 3. FIG. 3 illustrates an embodiment of the invention wherein the rim 16 is of square cross-section dimensioned as shown and with the rim exhibiting a mean radius R as illustrated. The modification according to FIG. 4 illustrates a rim 16' of circular cross-section.

The dissipation of energy by the oscillator is due to internal friction of the hairspring and to friction forces acting upon the balance. The friction torque due to these latter forces when the pivot axis is horizontal maybe expressed by the following equation;

where the first term, a, is a constant term independent of oscillatory speed and proportional to the weight P of the balance rim, the radii r of the pivots for the balance, and the coefficient of friction 1; acting on such pivots, Thus, a 1; rP,

the second term is a viscous damping term due to air friction drag on the oscillating balance and is therefore proportional to the instantaneous angular velocity (I: of the balance and the surface area S of the balance rim; and,

the third tenn is a quadratic term due to non-linear effects of air friction on the balance rim and arms and which also is proportional to the surface area S.

In Equation l c K R,,,4 p where:

R, is the mean radius of the balance rim A is the cross-sectional area of the balance rim p is the density of the balance material g is the acceleration due to gravity 1 is the coefficient of friction at the balance pivots r is the balance pivot radius p is the dimensional girth of the balance rim cross-section K, and K are constants determined experimentally.

By fixing the terms pm, and r, the above may be reduced as follows:

where K K and K are constants.

Since the terms A and p are both related to the dimensions of the balance rim cross-section, the three expressions may be expressed in terms of only two variables, R,,, and s if the shape of the rim cross-section if fixed and s represents some linear dimension of the rim cross-section. For example, if the rim is of circular cross-section and s is the radius of the section, then A =1r s and p 2 1r s. Similarly, for a square cross-section where s is the length of one side, A =s and p 4s. Further, for a rectangular rim cross-section where the ratio of the sides is fixed, i.e., the ratio of the radial thickness s to the height h is fixed, then, h ns, n being constant, A ns and p 2s (n+1 Thus, if the area A is expressed in the general term A C s where the constant C, depends on the cross-sectional shape of the rim, and the girth p is expressed in the general term p C where the constant C, also depends upon the cross-sectional shape of the rim, the expressions (2), (3), and (4) may be rewritten as follows:

As will be shown below, the instantaneous power required to impart an amplitude 41 to the balance at a frequency w is as follows;

(8) P (2/7r)aw-i-(b/2)w -l-( 4/3'rr)cwd By substituting (5 (6), and (7) into (8.), there results:

(9) P= K.,R,,,C ,.S tlxlz-l-K R ,"C,,So.v d "+K R,,,C,,Smq5 K -K being constants and C and C, being also constants as established by the cross-sectional shape of the rim, as above.

From (9), with the initial available power (at zero hours) P fixed and the frequency w, as well as the initial amplitude d), fixed, a direct relation between R and S is established, thus:

( 10) P K R,,,C,,S m,, +K,,R,,, C,,Sw,, +K R,,,C,Sw 23 3 0 0 which reduces to:

l l K, K R,,,s -i-K R,,, .s-l-K R s; K K of course being constants (K, P Equation l l establishes R,,, as a function of s, i.e., R,,,=f(s) in which the initial amplitude condition 4:, is satisfied.

Equation (1 1) may be further simplified to the form:

Since s must be which yields a single solution for s as a function of R,

Since the moment of inertia I of the balance is equal to the product of its mass m times the square of its mean radius R, I m R,,,, and its mass is equal to 2 1r R,,,p A where the area A=C s as before,

(13) I 21rR C sp Since C, and p have been fixed,

Equation (14) establishes the moment of inertia as a function of R, and s, but since s=f(R,,,), see Equation (12), has already been established on the basis of P and da, the moment of inertia may be expressed directly as a function of either R,, or s alone, i.e., I=F(R,,,)=F'(s). Consequently, for various values of s and of f(s) yielding the corresponding F(R,,.) or F (.s'), the moment of inertia I may be calculated versus either s or R,,,. In FIG. 1, several of these curves, for different densities of the balance material, are shown corresponding to the following fixed values:

X, P, ergs/sec.

w 51r radians/sec.

the balance rim being of square cross-section so that A=s and p=4s.

For each value of I in the curves of FIG. 1, there will be a corresponding different power consumption in the 24-hour period so that, after 24hours, a different instantaneous power P t? P, will be available. The values of B are also shown in FIG. 1.

Based upon the energy consumption the instantaneous available power may be derived on the basis of the Airy formula:

d/dt=d =w, sin (WP-mp, sin 0 From Equation (1),

(l6) |C(0)| =a+bw,, sin0 I +Cmda,, sin" 0. Substituting (16) into and calculating the loss of amplitude for a quarter of a period to avoid difficulty with signs:

1 2 (a bwdao 5 610 41 from which the logarithmic decrement may be written:

From the equation for energy consumption Since instantaneous power then - a 500K +5 10%? 5 Cw 0 l fl? Equation (8) of course, will be recognized as the generalized form of Equation l 7).

Returning to the discussion of FIG. 1, the B of the mainspring is used as the lett-hand cut-off point for the I versus R, curve and the maximum value of I to the right of this cut-off point yields the most eflicient balance dimensions producing a loss of amplitude within the prescribed limits. The term B P- /Pand, from the considerations given above and from FIG. 1, a mainspring with B=0.76. is chosen. If it is desired that the loss of amplitude be smaller, a mainspring with B 0.76 may be used.

The metals and alloys which may be used are for instance:

Gold alloy 18 K density 15 Gold alloy 14 K density 13,8 Silver and gold alloy density 1 l Tungsten alloy Platinum density 2L5 It is also possible to use sintered materials.

The following table gives, by way of example, the charac-' teristics of some balances of the invention, in particular cases:

Fre- 1 Po Ms! q y s en. s=h

erg/s cm) A/h on (1124)] cm (mm) (mm) Mans watch 8* 15 18000 254 221 52 12.7 .53 Man's watch 8 15 36000 254 221 10.5 8.5 .45 Ladys watch 5 l5 l8000 254 221 27 NA .42 Lady's watch 5 15 36000 254 22l 5.4 7.7 .37

In the above table, the frequency A/h refers -to alternances per hour whereas the symbol D refers to the outside diameter of the balance wheel.

What is claimed is:

1. In a timepiece of the type having an energy source providing an initial instantaneous available power P,, and a balance comprising a hub and a rim oscillated at an initial amplitude 1b, and frequency to by said energy source, the improvement comprising,

said rim being of circular cross-sectional shape so that its cross-sectional area A=1rs and the perimeter p=2rrs, where s is the radius of the cross-section, the mean radius R,,, of said n'm from its center of oscillation and said dimension s being of fixed relation to establish said initial amplitude dz, at said frequency or with said initial power P,, and the value of R,, and the corresponding value of s being fixed to provide that maximum value of moment of inertia of the balance which depletes the instantaneous power available from said energy source by an amount such that after 24 hours of operation with the balance in the vertical position the instantaneous available power P imparts an oscillatory amplitude at, to the balance wherein du -d) does not exceed about 40.

2. A timepiece with a balance-hairspring assembly exhibiting a moment of inertia (I), drive means having an initial instantaneous available power P, for oscillating said balance at a given frequency and with a given initial amplitude d), and having after 24 hours of operation an instantaneous available power P for oscillating said balance at said given frequency and with an amplitude said balance having a rim of fixed cross-sectional shape disposed at an average radius R, from said assembly at a given frequency m with a given initial amplitude 1b,, and having after 24 hours of operation an available power P for oscillating said assembly at said given frequency and at an amplitude 4: which is less than P,,, said balance having a rim of fixed cross-sectional shape such that its mean radius R,, from its center of oscillation and its cross-sectional area (A) are functionally related according to where C,,s A the cross-sectional area of said rim C ,,s perimeter of said rim cross-section s radial thickness of the rim K K and K are constants P and 41 are fixed as given above, and said mean radius R,, being of a value such that said moment of inertia (I) is substantially maximum for said related values of R and (A) which produce amplitudes 42 according to the following:

P24 K m z4+ M F Z+ 3 J IJ QB characterized in that the difierence in amplitudes da -b does not exceed about 40.

4. In a time piece of the type having an energy source providing an initial instantaneous available power P and a balance comprising a hub and a rim oscillated at an initial amplitude 1), and a frequency w by said energy source, the improvement comprising,

said rim being of square cross-sectional shape so that its cross-sectional area A=s and the perimeter 4s where s is the length of one side of said cross-section, the mean radius R,, of said rim from its center of oscillation and said dimension s being of fixed relation to establish said initial amplitude (b, at said frequency m with said initial power P and the value of R and the corresponding value of s being fixed to provide that maximum value of moment of inertia of the balance which depletes the instantaneous power available from said energy source by an amount such that after 24 hours of operation with the balance in the vertical position the instantaneous available power P imparts an oscillatory amplitude dmto the balance wherein,, does not exceed about 40. 

1. In a timepiece of the type having an energy source providing an initial instantaneous available power Po, and a balance comprising a hub and a rim oscillated at an initial amplitude phi o and frequency omega by said energy source, the improvement comprising, said rim being of circular cross-sectional shape so that its cross-sectional area A pi s2 and the perimeter p 2 pi s, where s is the radius of the cross-section, the mean radius Rm of said rim from its center of oscillation and said dimension s being of fixed relation to establish said initial amplitude phi o at said frequency omega with said initial power Po, and the value of Rm and the corresponding value of s being fixed to provide that maximum value of moment of inertia of the balance which depletes the instantaneous power available from said energy source by an amount such that after 24 hours of operation with the balance in the vertical position the instantaneous available power P24 imparts an oscillatory amplitude phi 24 to the balance wherein phi o- phi 24 does not exceed about 40* .
 2. A timepiece with a balance-hairspring assembly exhibiting a moment of inertia (I), drive means having an initial instantaneous available power Po for oscillating said balance at a given frequency and with a given initial amplitude phi o and having after 24 hours of operation an instantaneous available power P24 for oscillating said balance at said given frequency and with an amplitude phi 24 , said balance having a rim of fixed cross-sectional shape disposed at an average radius Rm from its center of oscillation and said average radius Rm and the cross-sectional area (A) of said rim being of fixed relation, said rim having its average radius Rm selected to provide a maximum value of (I) for the instantaneous available powers Po and P24, the initial amplitude phi o , characterized in that the difference in amplitude phi o- phi 24 does not exceed about 40* .
 3. A timepiece movement comprising in combination a balance-hairspring assembly exhibiting a moment of inertia (I) and drive means having an initial power Po for oscillating said assembly at a given frequency omega with a given initial amplitude phi o and having after 24 hours of operation an available power P24 for oscillating said assembly at said given frequency and at an amplitude 24 which is less than Po, said balance having a rim of fixed cross-sectional shape such that its mean radius Rm from its center of oscillation and its cross-sectional area (A) are functionally related according to Po K, RmCaS2o+K2Rm3CpS phi o2+K3Rm4CpS phi o3 where Cas2 A1 the cross-sectional area of said rim Cps perimeter of said rim cross-section s radial thickness of the rim K1, K2, and K3 are constants Po and phi o are fixed as given above, and sAid mean radius Rm being of a value such that said moment of inertia (I) is substantially maximum for said related values of Rm and (A) which produce amplitudes phi 24 according to the following: P24 K, Rm CaS2 phi 24+K2Rm3CpS phi o2+K3Rm4CpS phi 243 characterized in that the difference in amplitudes phi o- phi 24 does not exceed about 40* .
 4. In a time piece of the type having an energy source providing an initial instantaneous available power Po, and a balance comprising a hub and a rim oscillated at an initial amplitude phi o and a frequency omega by said energy source, the improvement comprising, said rim being of square cross-sectional shape so that its cross-sectional area A s2and the perimeter p 4s where s is the length of one side of said cross-section, the mean radius Rm of said rim from its center of oscillation and said dimension s being of fixed relation to establish said initial amplitude phi o at said frequency omega with said initial power Po, and the value of Rm and the corresponding value of s being fixed to provide that maximum value of moment of inertia of the balance which depletes the instantaneous power available from said energy source by an amount such that after 24 hours of operation with the balance in the vertical position the instantaneous available power P24 imparts an oscillatory amplitude phi 24to the balance wherein phi o- phi 24does not exceed about 40* . 